Poisson Equation in Sobolev Spaces

Transcription

1 Poisson Equation in Sobolev Spaces OcMountain Dayligt Time. 6, 011 Today we discuss te Poisson equation in Sobolev spaces. It s existence, uniqueness, and regularity. Weak Solution. u = f in, u = g on (1) Definition 1. u W 1, () is a weak solution of te Poisson equation if u = f, in, u= g, on () u v + f v = 0 v W 1, 0 (); u g W 1, 0 (). (3) Te weak formulation is advantageous in getting quick estimates. For example, wen g = 0, we ave u W 1, C f L (4) for some constant C. To see tis, note tat wen g = 0, u W 1, 0 can be used as a test function, wic gives u = fu f L u L. (5) Applying Poincaré inequality gives te desired estimate. Existence. Te direct metod sows te existence/uniqueness of te solution of PDEs by studying its variational formulation. We sketc tis approac by studying te Poisson equation wit zero boundary condition: u= f, u W 0 1, (). (6) We know tat any weak solution to tis problem is a minimizer of te functional I(u)= u + fu. (7) We would like to sow tat te minimizer exists. We apply te so-called direct metod : Assume f L. 1. Writing I(u) u ε u 1 f (8) 4ε and recalling te Poincaré s inequality, we see tat I(u) as finite infimum.. Let u n be suc tat I(u n ) ց inf u W0 1,I(u). We sow tat tere is a subsequence converging to some limit u W 0 1,. To see tis, note tat a uniform bound on I(u n ) implies a uniform bound on un, since I(u) u L u L f L u L C u L =( u L C) u L. (9) by Hölder s inequality and Poincaré s inequality. 3. Uniform boundedness of u n L implies tat u n is uniformly bounded in W 0 1, and tus as a weakly 1 converging subsequence, still denoted by u n. We denote te limit by u. 1. Te weak convergence is in W 1,. Recall tat a sequence {u n } in a Hilbert space H is weakly convergent wit weak limit u H if (u n,v) (u,v) for any v H. Furtermore, using compact embedding, we see tat wen u n converges to u weakly in W 1,, we can find a subsequence, still denoted u n, converging to u strongly in L, at te same time u n converges to u weakly in L. 1

2 Poisson Equation in Sobolev Spaces 4. Te convexity of te functional I(u) ten guarantees tat wic means u is a minimizer. Uniqueness. Let u, v be weak solutions to Let w =u v. Ten w is weak solution to I(u ) liminf I(u n)= nր inf u W 0 1, I(u) (10) u= f in, u = g on (11) w = 0 in, w = 0 on. (1) Note tat as w C, we cannot directly apply te maximum principle for armonic functions. However proving troug te definition of weak solution is as simple. Left as a problem. Interior Regularity. Our goal is te prove te following teorem, wic justifies te intuition tat u is twice more differentiable tan f. By interior regularity, we mean we do not deal wit boundary data, and terefore te L -norm of u is necessary in te RHS. Teorem. Let u W 1, () be a weak solution of u = f wit f L (). For any, we ave u W, ( ), and u W, ( ) C ( u L () + f L ()). (13) were te constant depends on te distance between and. Furtermore, u= f almost everywere in. Remark 3. Te difficulty in proving te teorem lies in te fact tat we ave to sow u W,. Once tis is known, te estimate is relatively easy to establis. 1. We first sow tat u L ( ) 17 δ u L () witout any extra assumptions. Let η(x) be a cut-off function defined by 1 x η(x) = 1 1 δ dist(x, ) 0 dist(x, ) δ 0 dist(x, )>δ and set te test function Some calculation yields Using Young s inequality + δ f L (). (14) (15) v = η u W 0 1, (). (16) η u + (η u) (u η)= η f u. (17) ab ε a + 1 ε b a, b R, ε > 0 (18) on te nd term on te LHS and on te RHS we ave η u 1 η u + u η + 1 δ η u + δ η f. (19) Moving te first term on te RHS to te left, we ave ( 16 u δ + 1 ) δ u + δ f. (0)

3 OcMountain Dayligt Time. 6, Note tat u = u + boundary terms. if we assume u W 3,. Tus using u as test function we obtain u L ( ) Proof. Let, wit dist(, ) δ/4, dist(, ) δ/4. Now coose η C 0 1 ( ) wit η = 1 on and η 8/δ, and set were Ten we ave Recalling We ave f L (). (1) v = η i u () i u(x)= u(x+e i) u(x) ( i u) v = i ( u) v = u ( i v) = f i v (3) f L () v L ( ). (4) v = η i u (5) ( i u) (η i u) f L () (η i u) L ( ). (6) Te terms can be expanded to obtain η ( i u) f L () (η i u) L ( ) (η i u) ( i u η) f L () (η i u) L ( ) + 1 η 4 i u + 8 η i u. (7) Tis gives 3 η 4 i u f L () (η i u) L ( ) + 8 η i u. (8) To proceed furter, we need to study te two terms 1 8 (η i u) L ( ) 1 8 (η i u) L ( ). We ave (η i u) L ( ) Were we ave used te fact tat η η. 8 η i u. We ave η i u (sup η ) and 8 η i u. ( (η )) i u L + η i u L (sup (η ) ) i u L + η i u L. (9) i u = (sup η ) i u L. (30) Tus we ave 1 η i u f L () (sup (η ) ) i u L + 8(sup η ) i u L. (31) Te following lemma ten guarantees te existence of u and also gives te desired estimate. Lemma. Let i u u(x +e i) u(x), 0 (3)

4 4 Poisson Equation in Sobolev Spaces wit e i being te it unit vector of R n. Let and < dist(, ). Ten Proof. 1. If u L () and tere is K < suc tat ten u W 1, ( ) and. Conversely, if u W 1, ( ), ten i u L ( ) wit i u L ( ) K (33) xi u L ( ) K. (34) i u L ( ) xi u L ( ). (35) 1. We first sow tat i u converges as distributions in D ( ) to te distributional derivative of u. Ceck ( i u) ϕ = u ( i ϕ ) u( xi ϕ) (36) by Lebesgue s dominated convergence teorem. We ave ( xi u)(ϕ) = lim ( i u) ϕ i u L ( ) ϕ L K ϕ L, ϕ C 0 ( ). (37) Now recall tat C 0 ( ) is dense in L ( ), ( xi u) can be identified wit a bounded linear operator on L, wic means it can be identified wit a function in L ( ).. Since C is dense in W 1,, we only need to consider te case wen u C W 1,. In tis case we ave i u(x)= 1 xi u(x 1,, x i 1, x i + s, x i+1,, x n ) ds. (38) 0 Tis gives i u(x) 1 xi u(x 1,, x i 1, x i +s, x i+1,, x n ) ds (39) 0 due to Hölder s inequality. Now integrate over and excange te order of integration on te RHS we obtain te result. Wit te elp of tis lemma (part b) ) we ave 1 η i u f L () (sup (η ) ) xi u L + 8(sup η ) xi u L. (40) wic is a uniform bound on i u L ( ) η i u. (41) Now part a) of te lemma yields xi u L ( ) and also te desired estimate. Wen f as better regularity, we can differentiate te equation first and obtain te following interior regularity result. Teorem 4. Let u W 1, () be a weak solution of u = f. If f W k, (), ten u W k+, ( ) for any, and Here te constant depends on d,, dist(, ). Boundary regularity.. Riesz representation teorem. u W k+, ( ) C ( u L () + f W k, ()). (4)

5 OcMountain Dayligt Time. 6, We consider te Poisson equation wit Diriclet boundary condition: u = f in ; u= g on (43) were g can be extended to a function on te wole. Our purpose is to establis te following result: Teorem 5. Let u be a weak solution wit u g W 0 1, (). If f W k, (), g W k+, (), and be of class C k+, ten and we ave te estimate Te constant C depends on. Proof. We only give an outline ere. u W k+, (), (44) u W k+, () C ( f W k, () + g W k+, ()). (45) 1. First note tat since g W k+, (), we can replace u by u g and reduce te problem to. We first establis W 1, bound: u= f, u W 0 1, (). (46) u W 1, C ( g W 1,+ f L ). (47) To see tis, use v =u g as te test function. We obtain u (u g) = f (u g) (48) terefore u u g + f (u g) 1 4 u + g + 1 ε f +ε u g. (49) Apply Poincaré s inequality to te last term and coosing ε to be small enoug, we obtain te desired estimate. 3. For any, we can estimate xix j u C ( u + f ) C ( g W 1, + f L ). Terefore it suffices to establis te desired estimate in a neigborood of te bondary. 4. We illustrate te basic idea by assuming part of te boundary is in x n =0. We try to sow te W, bound for u in a small alf-ball B R + B R {x n >0}. Note tat once tis is done, te boundary, wic is compact, can be covered by finitely many suc balls. First note tat xi u is well defined in B R + and belongs to L (B R + ). Now let η be a cut-off function in C 0 (B R ). For all j n, j ± u is well-defined and we can use te test function j (η j u) as we did wen proving te interior regularity, and obtain te desired bound for all xi x j u except xn x n u. Now notice tat te equation implies n 1 xn x n u = f i=1 and terefore tis term enjoys te same bound as oter double derivatives. xi x i u (50) 5. For general, we need to first cover by small balls, and ten do a cange of variable on eac of te balls to straigten tat part of te boundary. After doing tis, owever, te equation does not ave te simple form u= f (51) anymore and proving te estimate becomes as difficult as proving similar estimates for te general case. Remark 6. It turns out tat wen te boundary is smoot, one can actually extend te regularity to.

6 6 Poisson Equation in Sobolev Spaces Teorem. Let R n be a bounded domain of class C, and let g C ( ), f C (). Ten te Diriclet problem possesses a unique solution u wic is C ( ). u= f in ; u= g on, (5) Te key to te proof is te embedding W k,p () C m ( ) for 0 m < k d p. L p Regularity. Teorem 7. Let 1< p<, f L p (), and let w be te Newton potential of f. Ten w W,p (), w = f almost everywere in, and w L p () C(n, p) f L p (). (53) Using tis teorem, we can obtain te following interior regularity result. Teorem 8. Let u W 1, () be a weak solution of u = f, f L p (), 1< p <. Ten u W,p ( ) for any, and Here C = C(n, p,, ). u W,p ( ) C ( u L p () + f L p ()). (54)